New Class of Location Measures, the Guard Estimator and Connections to Multiscale Variance Stabilization

This article introduces a new class of location estimators by applying an existing
measure of location to a set of multiscale means computed on the order statistics of a data set.
Class members are shown to have interesting and desirable properties such as: the member using
the geometric mean is always sandwiched between the geometric and arithmetic means.
The member using the median reduces to a simple form, the guard estimator,
which is the median of three quantities: the sample mean and two guard values based on two or four order
statistics nearest the median. The guard estimator is unbiased, consistent (at least for symmetric distributions),
computable in linear time, does not require tuning parameters and simulations suggest that
it achieves high efficiency: almost matching the mean or median's better efficiency under different distributions.
Guard's finite sample breakdown point demonstrates that it is highly robust even for small samples
and matches the median's breakdown value asymptotically. Two examples exhibit the new location
measures in action: one provides confirmation of a robust approach to establishing whether
Shoshoni leather goods were designed to the `Golden Ratio' standard; the other compares
four functional measures of location for the Aberporth wind speed series.

The new class of location estimators is inspired by the member that uses the geometric mean which
arises naturally from a theoretical analysis of multiscale variance stabilization (MVS) techniques.
The article
introduces maximum likelihood approaches for MVS techniques for independently and
identically distributed data
and sheds theoretical light on MVS by presenting analytical formulae for their Jacobians, a key component
of the likelihood.
The MVS techniques are shown empirically to compare favourably to the well-known Box-Cox
transform, but do not dominate it.